Fraction Division & Reciprocals

What is a reciprocal?

A reciprocal is what you get when you flip a fraction upside down. The reciprocal of 3/4 is 4/3. The reciprocal of 5 is 1/5.

Multiply a number by its reciprocal and you always get 1. So 3/4 × 4/3 = 1.

That is the reason the flip works. Dividing by a number undoes multiplying by it, and the reciprocal is the number that undoes it.

What is the area model?

An area model draws fractions inside a square. Cut the square into equal columns and rows, then shade a rectangle. The shaded rectangle shows a fraction of a fraction.

If the square has 3 columns and 4 rows, a rectangle 2 columns wide and 3 rows tall covers 2/3 × 3/4 = 1/2 of the square. You can see it.

The same picture answers a division. If you know the area of a rectangle and one of its sides, the other side is the area divided by the known side. So if the area is 2/3 and the height is 3/4, the width is the answer to 2/3 ÷ 3/4.

What they're learning

The module fixes the area and the height, and your kid drags the width until the shaded area matches the target. When it matches, the width is the quotient.

The idea is that the quotient is a missing factor. It is the number that, times the divisor, gives the dividend. That is why the worksheet checks the answer by multiplying back.

• A reciprocal is the fraction flipped. 3/4 becomes 4/3.
• A number times its reciprocal is 1.
• Dividing by 3/4 is the same as multiplying by 4/3.
• The quotient is the missing side of a rectangle.

Aliases

• Reciprocal
• "Flip and multiply"
• Keep, change, flip
• Dividing fractions

Refresher

A reciprocal is the fraction flipped upside down. The reciprocal of 3/4 is 4/3, and the reciprocal of 5 is 1/5. Any number times its reciprocal is 1. Dividing by a fraction is the same as multiplying by its reciprocal.

Try this together

Two or three minutes at the table is enough.

1. Open the module. The area is 2/3 and the height is 3/4. Ask your kid to guess the width before dragging.
2. Drag the corner until the area matches the target. Read the width. It should be 8/9.
3. Check it together: what is 8/9 × 3/4? It is 2/3, the area you started with. That is the missing factor.
4. Flip the divisor: what is 2/3 × 4/3? It is also 8/9. Show that flipping gives the same answer as dividing.

If the dividend is bigger than the divisor, the answer is more than 1 and this picture runs out of room. The other lesson, Dividing Fractions, handles those.

Words you might hear

• Reciprocal: a fraction flipped upside down. The reciprocal of 3/4 is 4/3.
• Dividend: the amount you start with. In the module it is the area.
• Divisor: what you divide by. In the module it is the height.
• Missing factor: the number that, times the divisor, gives the dividend. It is the same as the quotient.

The flip and multiply rule is quick, but it is easy to forget which fraction to flip. The picture gives your kid a way to check.

Deeper dive

Start with a rectangle whose area is 2/3 and whose height is 3/4. The width is whatever number times 3/4 gives 2/3, so the width is the answer to 2/3 ÷ 3/4.

Try 8/9. Multiply it back: 8/9 × 3/4 = 2/3. Now flip the divisor instead: 2/3 × 4/3 = 8/9. Both give the same answer, which is why the flip rule works. Dividing asks for the missing factor.